一个近似的VTI介质声波方程

An approximate acoustic wave equation for VTI media

声学近似是提高各向异性介质准纵波数值模拟和偏移处理计算速度的有效方法之一.为了获得VTI(具有垂直对称轴的横向各向同性)介质声波方程,根据VTI介质的精确相速度表达式,用待定系数法简化其中的根号项,得到一个近似相速度公式,由此公式通过反Fourier变换推导出一个时间二阶、空间四阶偏微分方程.定量相速度计算表明,当Thomsen参数ε=δ(即相应介质为椭圆各向异性介质)时,由该方程所确定的相速度是精确的;当ε≠δ时,该方程所确定的相速度随ε和δ差值的增大逐渐增加.该方程所确定的最大相速度百分比相对误差当ε=0.1、δ=0.2时为0.13%,当ε=0.8、δ=0.3时为-1.65%.有限差分数值模拟算例表明该方程是一个纯准纵波方程,其数值模拟波场快照中没有准横波.

Acoustic approximation is an effective way to simplify modeling and migration of quasi- P waves in the case of anisotropy. To generate a new acoustic wave equation for VTI media （transversely isotropic media with a vertical symmetric axis） from the accurate phase velocity expression, we make simplification to the square root term by an undetermined coefficients method to get an approximate phase velocity formula, from which we derive a second-order temporal and fourth-order spatial partial derivative equation through inverse Fourier transform. Quantitative computations show that the phase velocity specified by the new acoustic equation is accurate when Thomsen parameter ε is equal to δ （elliptical anisotropy） and the errors of the phase velocity become bigger as the difference between ε and δ gets bigger. The maximum relative error of the phase velocity is 0.13% in the case of ε=0.1 and δ=0.2, and becomes -1.65% in the case of ε= 0.8 and δ= 0.3. The shear-wave-free snap shots from finite difference modeling demonstrate that the new acoustic equation is a pure quasi-P wave equation.